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The position of a particle is given by $\overrightarrow{r}=3t\hat{i}+\sqrt{3}{t}^2\hat{j}-4\hat{k}$, where $t$ is in seconds and $\overrightarrow{r}$ is in meters. Find out magnitude and direction of velocity $\overrightarrow{v}$ with horizontal at $t=\sqrt{3}\text{s}$.

A circular disc is rotating with constant $\omega =2$ rad/sec, about an axis passing through its centre. An object is kept at a distance of ${}^{\prime }{r}_1=2c{m}^{\prime }$ from the centre and another object is kept at a distance of ${}^{\prime }{r}_2=1c{m}^{\prime }$. Find the ratio of their linear speeds.

Which represents NAND gate

A body is projected along a rough horizontal surface with a velocity $6\text{m/s}$. If the body comes to rest after travelling $9\text{m}$, then coefficient of sliding friction, is $(g=10{\text{m/s}}^2)$

A spherical body of radius $b=\frac{4}{\pi }$ $\text{m}$ has a concentric cavity of radius $a=\frac{2}{\pi }$ $\text{m}$ as shown in the figure. Thermal conductivity of the material is $100{\text{W/m}}^{\circ }\text{C}$. If temperature of inner surface is kept at ${70}^{\circ }\text{C}$ and of the outer surface at ${30}^{\circ }\text{C}$, then find the rate of heat flow. (in $\text{kW}$)

A mass of block $8\text{kg}$ is attached by a rope with other block of mass $4\text{kg}$ as shown in the figure. Find the acceleration of COM of the system.

(Neglect friction everywhere and neglect the mass of the rope and pulley)

The figure shows a compressed light spring and block system kept on a horizontal frictionless plane. Which of the following graph shows the correct relation between acceleration $\left(a\right)$ of block and distance $\left(x\right)$ moved by it when released till spring come to its natural state?

Two spheres made of the same material have the same diameter. One sphere is hollow and the other is solid. If they are heated through the same range of temperature,

A thin square plate of side $4\text{cm}$ and mass $m$ has a moment of inertia $I_{\text{Sq}}$ about the line $AB$ as shown in figure. It is then reformed into a rectangle of length $8\text{cm}$ and breadth $2\text{cm}$ with a moment of inertia $I_{\text{Rect}}$ about the line $CD$ as shown in the figure. What is the value of the ratio $\frac{I_{\text{Sq}}}{I_{\text{Rect}}}?$

A cylinder of mass $M$ has a length $L$ that is $\sqrt{3}$ times its radius $R.$ What is the ratio of its moment of inertia about its own axis and that about an axis passing through its centre and perpendicular to its axis?

The fundamental frequency of a sonometer wire increases by $6\text{Hz}$ if the tension in the wire is increased by $44\%$, keeping the length constant. What is the change in the fundamental frequency of the wire when the length is increased by $20\%$, keeping the original tension in the wire constant.

A construction worker's helmet slips and falls when he is $78.4\text{m}$ above the ground. He hears the sound of the helmet hitting the ground $4.23$ seconds after it slipped. Find the speed of sound (in SI units) in air.

Find the incorrect option for the general rotation of a wheel of radius $r$ which can be viewed as pure translation with velocity $\overrightarrow{V_0}$ and pure rotation about $\text{O}$ with $\overrightarrow{\omega }$.

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two vectors perpendicular to each other and $|\overrightarrow{a}|=|\overrightarrow{b}|=1$ . Determine the angle between $\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{2a}+\overrightarrow{b}$

What are the dimensions of K (coulomb’s law constant)?

The capacity of a condenser is 4x10^-6 farad and its potential is100 volts. The energy released on discharging it fully will be

A uniform rope of mass $0.1\text{kg}$ and length $2.5\text{m}$ hangs from ceiling. The speed of transverse wave in the rope at upper end and at a point $0.5\text{m}$ distance from lower end will be:

The sound level at a point is increased by 30 dB. By what factor is the pressure amplitude increased?

Add vectors A & D. Which of these is the correct result?

A solid body rotates about a stationary axis with an angular deceleration $\beta \propto \sqrt{\omega }$ where $\omega$ is its angular velocity. If at the initial moment of time, its angular velocity was equal to ${\omega }_0$, then the mean angular velocity of the body averaged over the whole time of rotation till it comes to rest is,

In the given figure , $a=10\sqrt{2}{\text{m/s}}^2$ represents the total acceleration of a particle moving in the clockwise direction on a circle of radius $R=2\text{m}$ at a given instant of time. Speed of the particle is:

Find the magnetic moment of a rotating rod of mass M, charge Q and length L rotating about an end point at $\omega rad{s}^{-1}$

If a particle has velocity $v\left(t\right)=18-6t\text{m/s}$, the magnitude of average velocity in $10\text{sec}$ is

The minimum wavelength of X-rays produced in an X-ray tube is $\lambda$ when the operating voltage is V. What is the minimum wavelength of the X-rays when the operating voltage is $\frac{V}{2}$?

A hollow sphere of mass $m$ = 1 kg and radius $R$ = 1 m rests on a smooth horizontal surface. A simple pendulum having string of length $R$ and bob of mass $m$ = 1 kg hangs from top most point of the sphere as shown in the figure. A bullet of mass $m$ = 1 kg and velocity $v$ = 2 m/s partially penetrates the left side of the sphere. The velocity of the sphere just after collision with bullet is:

A train accelerates from $36\text{km/h}$ to $54\text{km/h}$ in $5\text{s}$ then average acceleration of the train is

A ring of radius $5\text{m}$ and linear mass density $\lambda =\frac{0.1}{\pi }\text{kg/m}$ is spinning about an axis passing through its COM perpendicular to its plane. Find the moment of inertia of the ring about this axis.

Two blocks $A$ and $B$ of same mass $m$ connected by a light spring are suspended by an inextensible string as shown in the figure. Find the magnitude of the acceleration of block $A$ and $B$ just after the string is cut.

A person trying to lose weight by burning fat lifts a mass of $10\text{kg}$ upto a height of $1\text{m}$ for $1000$ times. Assume that the potential energy lost each time he lowers the mass is dissipated. How much fat will he use up, considering work is done only when the weight is lifted up? Fat supplies $3.8\times {10}^7\text{J/kg}$ which is converted to mechanical energy at an efficiency of $20$% . Take $g=9.8{\text{ms}}^{-2}$. Consider weight to be lifted slowly.

Two forces $\overrightarrow{F_1}=2\hat{i}-5\hat{j}-6\hat{k}$ and $\overrightarrow{F_2}=-\hat{i}+2\hat{j}-\hat{k}$ are acting on a body at a point $(1,1,0)$ and $(0,1,2)$ respectively. Find the torque acting on the body about the point $(-1,0,1)$.

A particle is resting on a smooth horizontal floor. At $t=0$, a horizontal force starts acting on it. The magnitude of the force increases with time according to law $F=\alpha t$, where $\alpha$ is a constant. For the figure shown which of the following statement is wrong?

An ant starts crawling from the bottom of a bowl of radius $r$, having coefficient of friction $\mu$. The maximum vertical height $h$ up to which it can reach is

A parallel-plate capacitor with plate area A and separation between the plates d, is charged by a constant current i. Consider a plane surface of area A/2 parallel to the plates and drawn symmetrically between the plates. Find the displacement current through this area.

A total charge $Q$ is broken in two parts $Q_1$ and $Q_2$ and they are placed at a distance $R$ from each other. The maximum force of repulsion between them will occur, when

A motorcycle is moving with speed $20\text{m/s}$ on the road. The M.O.I of wheel about its axis of rotation is $2{\text{kg m}}^2$ and radius of its wheel is $0.5\text{m}$. Motorcycle comes to rest in $10\text{sec}$ after applying the brake. Find the magnitude of the average torque transmitted by its brake to the wheel?

Two non-mixing liquids of densities $1000{\text{kg/m}}^3$ and $2000{\text{kg/m}}^3$ having height of $1\text{m}$ each are taken in a container, as shown in the figure. When a solid cylinder of length $0.5\text{m}$ and density $\rho$ is inserted in this container, it is observed that the cylinder floats with its axis being vertical and a length $0.2\text{m}$ of it in the denser liquid. The density $\rho$ of the solid cyclinder will be equal to

A pulley of $10cm$ diameter is wrapped with a cord of length $2m$ around its periphery. If pulley is initially at rest and is given an angular acceleration of $0.5rad/s^2$, then total angular displacement of the pulley for the cord to unwind completely and the time taken by the cord to unwind completely are

From a tower of height $H$, a particle is thrown vertically upward with a speed $u$. The time taken by the particle, to hit the ground, is $n$ times that taken by it to reach the highest point of its path. The relation between $H,u$ and $n$ is

In the arrangement shown in the figure, if $v_1$ and $v_2$ are instantaneous velocities of masses $M_1$ and $M_2$ respectively, and angle $ACB=2\theta$ at the instant, then

When a spiral spring is stretched by suspending a load on it, the strain produced is called

Match column A with their correct differentials in column B



$$\begin{array}{cccccc}& ColumnA& & ColumnB& & \\ p.& cosx& \left(i\right)& sinx& \left(v\right)& si{n}^{2}x\\ q.& tanx& \left(ii\right)& cosx& \left(vi\right)& se{c}^{2}x\\ r.& cosecx& \left(ii\right)& sinxcosx& \left(vii\right)& -cotxcosecx\\ s.& sinx& \left(iii\right)& tanxsecx& \left(viii\right)& cotxcosecx\\ t.& cotx& \left(iv\right)& -sinx& \left(ix\right)& -cose{c}^{2}x\end{array}$$​

Nuclear forces are

A particle of mass $m$ starts moving in a circular path $A$ of constant radius $r$ in such a way that its centripetal acceleration $a_c$ is changing with time as $a_c=k^2{r}^2{t}^2$ where $k$ is a constant. What is the power delivered to the particle by the force acting on it?

A block of mass 8 kg is placed on a rough inclined plane and subjected to a force of 30 N directed up the plane as shown in the figure. If the inclination of the plane is 60° and the coefficient of static friction between the block and the plane is 0.4, then total contact force exerted by the block on the plane can be directed along



8 kg द्रव्यमान के एक गुटके को खुरदरे आनत तल पर रखा जाता है तथा इस पर चित्र में दर्शाए अनुसार तल के ऊपर की ओर निर्देशित 30 N का बल आरोपित किया जाता है। यदि तल का आनति कोण 60° है तथा गुटके व तल के मध्य स्थैतिक घर्षण गुणांक 0.4 है, तब गुटके द्वारा तल पर आरोपित कुल सम्पर्क बल किसके अनुदिश निर्देशित हो सकता है?

A ball is swinging with the help of thread as shown in the figure. The component of $mg$ perpendicular to thread is

A body is moving unidirectionally under the influence of a source of constant power, its displacement in time $t$ is proportional to

What will be the total resistance offered to the flow of water by the system of pipe $1$, pipe $2$, pipe $3$ and pipe ignoring the joints? If all the pipes are of length $3\text{m}$ and radius $2\text{m}$ and '$\eta$' is viscosity of water. (Assume $\pi =3$)